How to Incorporate Step-Wise Potential into Schrödinger Equation for a 1D Box?

[Litmus]

Homework Statement

Trying to construct Shrodinger Equation given:
* mass: m

* Boundary Conditions: (potential)
V(x)=-Vo exp(-x/L) for 0<x≤L
V(x)=∞ for x≤0

Homework Equations

The Attempt at a Solution

(-h^2 / 2m ) (d^2 ψ / dx^2) + V(x)ψ = E * psi

Not sure how to incorporate step-wise V(x) into above eq.

 

[DrClaude]

Hi Litmus, welcome to PF!

* Boundary Conditions: (potential)
V(x)=-Vo exp(-x/L) for 0<x≤L
V(x)=∞ for x≤0

What about $x> L$?

And what can you say about the wave function for $x < 0$?

 

[vela]

You need to solve the Schrödinger equation in each region and then match the solutions at the boundaries. Your book should have examples of how to do this.

 

[Litmus]

What about x>L?

And what can you say about the wave function for x<0?

I don't understand. I've given conditions x<0, and x>L it's not specified. Why do we care?

You need to solve the Schrödinger equation in each region and then match the solutions at the boundaries. Your book should have examples of how to do this.

Like this?
(-h^2 / 2m ) (d^2 ψ / dx^2) + [-Vo exp(-x/L) ] ψ = E * psi
(-h^2 / 2m ) (d^2 ψ / dx^2) + ∞ψ = E * psi

Boundary is 0 so:
(-h^2 / 2m ) (d^2 ψ / dx^2) + [-Vo] ψ = E * psi

... how do I proceed?

 

[unscientific]

I don't understand. I've given conditions x<0, and x>L it's not specified. Why do we care?



Like this?
(-h^2 / 2m ) (d^2 ψ / dx^2) + [-Vo exp(-x/L) ] ψ = E * psi
(-h^2 / 2m ) (d^2 ψ / dx^2) + ∞ψ = E * psi

Boundary is 0 so:
(-h^2 / 2m ) (d^2 ψ / dx^2) + [-Vo] ψ = E * psi

... how do I proceed?

The problem can be tackled in the following steps:

1. Use TISE to get a general form of the wavefunction
2. Solve for the constants in the general wavefunction using boundary conditions